On-Line Parameter Estimation in General State-Space Models

The estimation of static parameters in general non-linear non-Gaussian state-space models is a long-standing problem. This is despite the advent of Sequential Monte Carlo (SMC, aka particle filters) methods, which provide very good approximations to the optimal filter under weak assumptions. Several algorithms based on SMC have been proposed in the past 10 years to solve the static parameter problem. However all the algorithms we are aware of suffer from the so-called `degeneracy problem'. We propose here a methodology for point estimation of static parameters which does not suffer from this problem. Our methods take advantage of the fact that many state space models of interest are ergodic and stationary: this allows us to propose contrast functions for the static parameter which can be consistently estimated and optimised using simulation-based methods. Several types of contrast functions are possible but we focus here on the average of a so-called pseudo-likelihood which we maximize using on-line Expectation-Maximization type algorithms. In its basic form the algorithm requires the expression of the invariant distribution of the underlying state process. When the invariant distribution is unknown, we present an alternative which relies on indirect inference techniques.